Sequences and Series — Textbook Review, Convergence Tests & Writing Solutions

Based on the supplied transcript. Punctuation and paragraph breaks have been added, with repeated fillers and false starts reduced for readability. Obvious transcription errors have been corrected. The speaker’s opinions, recollections, and descriptions of course requirements are retained.

All right. [Coughs and clears throat.] Let’s find a calculus book. See if we can find a Stewart Calculus PDF. Let me get a file here. We’re going to do sequences and series.

There’s a book. That’s the one I want. That’s the girl I love.

I’m looking for this book, and I’m going to have ChatGPT cite it quickly. I’ll start this as my sequences and series crash course.

The textbook used as a reference will be this book. We’ll review it. I found it here, located through that. Let’s start the Sequences and Series folder.

Let me get the ball rolling. Mic check. Where’s my snare?

I’m going to do a sequences and series crash course. We’ll look at Stewart’s calculus book first. This is for The STEM Majors Guide to the University.

Let me get my cover. I thought I had it already. Was it in the other folder? I want to use new book covers.

I don’t know whether I made a series cover yet. Actually, this will go under calculus. Let me find that—not precalculus. Apparently, I have two precalculus covers and no calculus. There it is.

This will go under the calculus file on the YouTube channel and my website. I’ll add it before we look at the book so everything is organized and you know where to find it.

I have The Ultimate Crash Course series running alongside Math Monthly. What did I label calculus? Number six? Bear with me, children. I want everything in order before solving problems.

The crash courses operate differently. I’m going to solve problems rather than write a book. The Math Monthly books are for publication. The crash courses are for review.

If you’re taking the course and watching this, you’re cheating and plagiarizing. That’s your choice, but I have to warn you.

We’ll go to Chapter 11 and review sequences and series in Stewart’s calculus book. Then I’ll solve problems. This is an individual video to begin the playlist.

Let me do one more thing. I’ll review the book briefly and point out a few things before the crash course.

PNG download—that’s for my podcast playlist on YouTube. Let me organize it: Calculus, The Ultimate Crash Course for PLEM Majors.

I know you’re thinking, “Don’t you keep telling us not to watch this?” It isn’t for while you’re in the course. It’s fine before or after.

I’d be a hypocrite if I said I didn’t do it myself. That’s when I realized how bad it was for my education.

I nearly failed calculus because I watched PatrickJMT do sequences and series. I went to the professor and said, “I got every answer right. What happened?”

He said, “Who cares about the answers? Your solutions were garbage.” [Laughter.] That’s when it began to click. I’m trying to communicate what clicked for me.

This will be the first video in the library. I’ll do the sequences and series crash course now, probably a question from each section over the next hour, as individual videos.

This is being recorded live on Zoom. If you were a Math Monthly member, you could join and ask questions. This is separate from the journal; it’s for the crash course series.

Sequences and series are essentially the same in Thomas’ Calculus. Other books are similar, though not as uniform. Many universities use this particular book.

In a couple of weeks, many students will be searching for sequences and series. For most, it’s the hardest part of calculus. I thought it was the easiest; I just didn’t write the solutions correctly because I watched someone online solve them with chicken scratch.

First, a sequence is a set of objects, such as a equals a-one, a-two, and so on. Second, a series is the sum of those objects: a-one plus a-two and so on. You can call that S.

This is English. If you don’t edit grammar and punctuation, you aren’t doing mathematics; you’re a child playing with puzzles. You’re writing an English paper every time.

At this level, writing English should be clear to you. You had 18 years of school to learn it.

There’s one section on sequences. Then you move to series and a collection of tests.

Each test has a theorem and definitions. You apply them. If you don’t follow the theorem and definitions exactly, you’ll fail the test.

That’s where students go wrong when using the internet, AI, ChatGPT, Khan Academy, or tutors instead of the assigned book. You’re playing with puzzles instead of doing mathematics.

I nearly failed doing the same thing. This was the point where I stopped watching the internet.

Here’s a theorem. Many of you have been scared to read these, but now you can’t solve the problems without them. If you haven’t learned to read them, that becomes a problem. That’s what I’ll show tonight.

You’ll have geometric series, alternating series, the integral test, ratio test, root test, and limit tests.

There are two—no, I’m thinking of the comparison tests. There’s limit comparison and direct comparison, not two limit tests. I haven’t done this in about a year.

You need to write the theorem on the exam to receive credit. That’s what happened to me. I did the arithmetic but didn’t address the theorem.

The professor said, “Jonathan, you got the answer right, but you didn’t cite the source.” I asked what he meant. That’s when it began to click.

Not writing the theorem is like discussing data in a scholarly paper without references. In college, you’re writing scholarly papers.

You may not have connected that every time you write a mathematical equation, you’re writing an English essay. You aren’t a child playing puzzles.

Some ask why nobody told them. You spent years writing English. What do you think this is: the integral of f of x, dx, equals one, followed by a period?

Is that a mathematical equation or an English sentence? “The integral of the function evaluated at x, differential x, is equal to one.” It’s a condensed sentence.

If you haven’t made that connection, I’m here to help. I want you to succeed, but you have to put down your intellectual sword.

There aren’t arguments in mathematics in the sense of arguing with people. There are arguments about whether something works, which you prove.

Arguing with someone about mathematics is like watching a blind person try to teach a deaf person. It won’t work well.

There isn’t much else to review here. I want to emphasize that, on an exam, if you don’t identify the test you’re using, you may receive no credit or lose half.

That happened to me. My answers and arithmetic were right, but I didn’t say, “By the limit comparison test, under these conditions, this follows.” I just did arithmetic.

That’s when it came together for me. You get the point.

This video serves a couple of purposes. One is to cite where I found the book so the publisher can see it’s available online and request removal.

We’re training to do things correctly. If you study mathematics, physics, or engineering, you’re a mathematician, physicist, or engineer by definition. There isn’t an intellectual threshold for using the name.

If you study it, you accept a code of honor, truth, and following rules.

I found the book here. This site probably doesn’t have the right to distribute it freely, so I make the citation clear so the publisher can request a cease-and-desist or removal.

It’s also for education. Whether I have the right to use it is uncertain, but that’s part of why I identify the source.

I just had a big soda, and the sugar is kicking in. My face is turning red. I had soda and cake because I’m having two implants put in tomorrow, so I had a giant meal.

Another thing about this chapter: if you haven’t reached it, they may give you the entire chapter in the last week or two—about 11 sections.

It looks frightening, but you’re doing integration and limits, what you’ve already been doing. Now you apply that arithmetic to theorems.

As you progress to junior- and senior-level work, you apply the theory. What use can you find for it?

I once tutored a student on calculating areas of an ever-changing shape, like cutting a seashell down the middle. We created a series, summed it, and used that to find what we needed.

I don’t remember the exact problem. It was years ago. Let me see whether there’s something similar here.

It wasn’t necessarily in this book. She was at a university in Los Angeles or something. I don’t remember.

Here’s a similar sort of question. There are applications. Doing theoretical problems is one thing; finding a use is another.

That’s the issue with my discovery—or not an issue; I haven’t gotten around to it. I submitted an original discovery to the Journal of Number Theory, and they said, “Find applications, and we’ll publish it.”

I said, “I’ll get around to it.” [Laughter.] Those who join the research program may see it someday. We’ll work on the discoveries together.

This is a brief review, an explanation of where I found the book, and preparation of the playlist and crash course.

These solutions won’t be written as formally as the journal solutions. I’m going to solve them and show how they’re done.

These crash courses are for before or after a course. You can watch before taking it, or review later in senior year or graduate school.

You misunderstand when I say that, during the course, the syllabus says to use the textbook. I didn’t say never watch videos. I said not to leave the assigned material while taking the course.

It defeats the purpose. If you aren’t up for the challenge, don’t take it on.

We’ll end here. Let me upload this, and then we’ll start the first—[recording ends].

Author Jonathan David introduces a sequences and series review using a Stewart calculus textbook. The video prepares the playlist, previews the chapter, and explains the importance of supporting convergence conclusions with the appropriate definitions and theorems.

Sequences vs. Series

The introduction distinguishes a sequence of terms from a series formed by adding those terms. More precisely, a sequence is ordered and indexed, and convergence of an infinite series is determined by the limit of its partial sums.

Choosing & Applying a Test

The discussion previews geometric series, alternating series, the integral test, direct comparison, limit comparison, the ratio test, and the root test. Jonathan emphasizes reading the relevant theorem and checking its conditions rather than treating the calculation alone as a complete justification.

Explaining the Conclusion

Jonathan recalls losing credit despite obtaining correct answers because his written work did not identify or justify the method used. He uses this experience to explain the role of clear sentences, named tests, hypotheses, and conclusions in mathematical writing. Exact grading expectations depend on the instructor.

Familiar Calculations in a New Setting

Limits and integration remain important tools in this chapter. The new task is to connect those calculations to convergence theorems and explain what the calculations establish.

Theory & Applications

The video briefly discusses using series in an applied area problem and distinguishes establishing a theoretical result from identifying applications. Jonathan also mentions a personal research project he hopes to revisit with the group.

Crash Course vs. Journal

The crash course is described as a collection of individual problem-solving videos for preparation or later review. Jonathan distinguishes these recordings from the more formally written material developed for the Math Monthly journal.

The supplied transcript does not provide the textbook’s complete edition details or source URL, so no bibliographic citation is invented here. Course rules determine permitted resources; a source citation alone does not establish permission to reproduce a textbook.